Deal all,
Thank you very much for the quick replies — you are right that my previous
description lacked important details. I’ll summarize the setup more
precisely and then show the numerical behaviour.
*Problem / setup:*
-
I follow the paper *“Multigrid methods for H(div)-conforming
discontinuous Galerkin methods for the Stokes equations”*.
-
Finite elements: RT_k-DGQ_k (so an H(div)-conforming RT velocity, DG
pressure).
-
Solver: FGMRES for the global linear system; multigrid used as a
preconditioner; the multigrid smoother is a multilevel Schwarz method.
-
PDE form: I use the *stress-tensor* form of the Stokes equations (same
form as in deal.II step-22). But I think under the divergence-free
condition these two forms should be equivalent.
-
Boundary conditions: I implement inhomogeneous Dirichlet boundary
conditions. Nitsche’s method is applied in the weak form for enforcing the
velocity BCs. In setup_system() I also apply
`` VectorTools::project_boundary_values_div_conforming(...) `` to the
velocity block.
-
Pressure: I did *not* apply any extra constraint or projection in the
weak form — the pressure block is assembled directly from the weak
formulation (no extra stabilization there).
*About the pressure constant / mean value:*
Before computing the error I tried to remove the mean pressure via
const double mean_pressure = VectorTools::compute_mean_value( dof_handler,
quadrature_over_integration, solution, dim);
solution.block(1).add(-mean_pressure);
However ``mean_pressure`` turned out to be very small (at least much
smaller than the pressure error I observe), so this subtraction did not
practically change the results.
*Observed behaviour: (The* penalty parameters *follow the choice in *
step-74)
-
With *homogeneous Dirichlet* BCs everything behaves as expected
(convergence rates ok).
-
With *inhomogeneous Dirichlet* BCs the *pressure L2 error does not
converge* (velocity behaves better only in RT1)
======================= RT1-DGQ1 ========================
cells uL2 uH1
pL2
4 2.067e-01 - 3.128e+00 - 2.055e+00 -
16 5.324e-02 1.96 1.596e+00 0.97 4.447e-01 2.21
64 1. 307e-02 2.03 7.968e-01 1.00 1.170e-01 1.93
256 3.256e-03 2.01 3.983e-01 1.00 4.811e-02 1.28
1024 8.195e-04 1.99 1.997e-01 1.00 2.965e-02 0.70
4096 2.086e-04 1.97 1.005e-01 0.99 2.049e-02 0.53
16384 5.409e-05 1.95 5.090e-02 0.98 1.444e-02 0.51
======================= RT2-DGQ2 ==========================
cells uL2 uH1 pL2
4 3.405e-02 - 6.559e-01 - 4.237e-01 -
16 4.760e-03 2.84 1.666e-01 1.98 6.974e-02 2.60
64 7.869e-04 2.60 5.200e-02 1.68 4.539e-02 0.62
256 2.285e-04 1.78 2.778e-02 0.90 4.671e-02 -0.04
1024 8.213e-05 1.48 1.923e-02 0.53 3.686e-02 0.34
4096 2.966e-05 1.47 1.371e-02 0.49 2.707e-02 0.45
16384 1.061e-05 1.48 9.751e-03 0.49 1.944e-02 0.48
If helpful I can post minimal code snippets (the weak form assembly, the
boundary projection call, and the lines where I subtract mean_pressure) and
the convergence tables/logs. Thank you again for any guidance.
Best regards,
HC
On Wednesday, 1 October 2025 at 09:50:27 UTC+2 [email protected] wrote:
> In addition to what Wolfgang mentioned, I was wondering how you were
> managing the pressure constant in this case. This may severely hinder your
> "apparent" convergence, especially for pressure.
>
> On Wednesday, October 1, 2025 at 1:26:35 a.m. UTC+2 Wolfgang Bangerth
> wrote:
>
>>
>> On 9/30/25 13:38, HC Zhang wrote:
>> > I am solving the Stokes equations using Raviart–Thomas elements in
>> > deal.II. When the boundary conditions are homogeneous Dirichlet,
>> > everything works fine and I observe the expected convergence behavior.
>> >
>> > However, when I switch to inhomogeneous Dirichlet boundary conditions,
>> I
>> > encounter problems: although I apply
>> > ``VectorTools::project_boundary_values_div_conforming(...)`` to set the
>> > boundary values, the pressure error does not converge.
>> >
>> > Has anyone experienced a similar issue, or could point me to the right
>> > way of handling inhomogeneous Dirichlet conditions for the Stokes
>> > problem with RT elements? Am I missing an additional step to make the
>> > pressure converge?
>> >
>> > Any advice or references would be very much appreciated.
>>
>> HC: I think there isn't enough information here. For example, I'm not
>> even sure what variable you're providing boundary values.
>>
>> Regardless, what have you tried already? When you compare the computed
>> and the exact solution, which variable differs? The velocity or the
>> pressure? If you look at a visualization of the solution, in which
>> specific ways does the computed solution differ from the expected one?
>> Is it offset by a constant? Are the boundary values completely wrong?
>> For example, do you expect the boundary values to be g(x) but they are
>> zero? When you say "the pressure error does not converge", does that
>> mean that it doesn't converge to the right value, or that it in fact
>> *diverges*?
>>
>> I often find it useful to be specific in which ways the solution does
>> not match my expectations when trying to find errors.
>>
>> Best
>> W.
>>
>>
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